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Quantum Physics

Title:
Holonomy for Quantum Channels

Abstract: A quantum holonomy reflects the curvature of some underlying structure of
quantum mechanical systems, such as that associated with quantum states. Here,
we extend the notion of holonomy to families of quantum channels, i.e., trace
preserving completely positive maps. By the use of the Jamio{\l}kowski
isomorphism, we show that the proposed channel holonomy is related to the
Uhlmann holonomy. The general theory is illustrated for specific examples. We
put forward a physical realization of the channel holonomy in terms of
interferometry. This enables us to identify a gauge invariant physical object
that directly relates to the channel holonomy. Parallel transport condition and
concomitant gauge structure are delineated in the case of smoothly parametrized
families of channels. Finally, we point out that interferometer tests that have
been carried out in the past to confirm the $4\pi$ rotation symmetry of the
neutron spin, can be viewed as early experimental realizations of the channel
holonomy.